is a great resource for Austin and Hill Country fly fishing in general as is the book Fly Fishing the Texas Hill Country by Kevin Hutchinson.
The fishing is incredible in Austin, there is an abundance of public land and great access to water. If there is meaningful water, there will be fish there. You'd be shocked at some of the places in the middle of town where you can catch amazing fish. Barton Creek is incredible fishing throughout although there is a steep learning curve since the water is so clear. I was out by Lost Creek last Thursday and managed to trick this chunky Guadalupe Bass into taking my fly last Thursday.
Bull Creek District Park on Lakewood is my go to spot for sunfish action with a two weight. I have caught some great bass and sunfish in Shoal Creek behind the restaurant Shoal Creek Saloon. St. Edwards Park up Spicewood Springs Road has some very giant bass in the deep holes if you can trick them. Brushy Creek in Round Rock has a lot of nice fish but keep your eyes open for snakes as well. If you've got time, head south and fish the San Marcos, Blanco, and Guadalupe rivers, there are some good public access points and since it is a tailwater the Guadalupe fishes very well in the summer for bass but you've got to get up early to beat the tubers.
I would really urge you to get out and just start exploring, the water around Austin really makes it very special and the fishing can't be beat.